Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418500 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1003 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,6)(7,8)(9,10)(11,14)(12,16)(13,15)(17,18,23,28,24,29,22,27)(19,26,30,25,31,32,21,20);(1,2,5,6)(3,7,4,8)(9,11,13,12)(10,14,15,16)(17,19,24,31)(18,25,29,20)(21,22,30,23)(26,27,32,28);(1,3,5,4)(2,7,6,8)(9,12,13,11)(10,14,15,16)(17,20,22,32,24,25,23,26)(18,19,27,21,29,31,28,30);(1,4,5,3)(2,7,6,8)(17,21,22,31,24,30,23,19)(18,26,27,20,29,32,28,25)Polycyclic presentation:
2^9|a^2=g,b^2=efgi,c^2=ef,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 348979);